**Introduction to growth and decay factor:**

Exponential growth is nothing but the exponential function occurs if the rate of growth is proportional to the functions current value in themathematical functional part. In discrete domain it contains some equalintervals where it can be called as the geometric growth or geometric decay factor. The geometric factor value forms the geometric progression. Now we are going to see about the exponential growth and the decay factor.

Find the exponential growth if it takes $1200 to double at 21/2 % compounded continuously

**Sol:**

First we have to take the formula of growth factor,

A = Pe^{rt}

The rate value which can be taken as 0.105

2400 = 1200 e^{ 0.105t}

We have to take natural log on both sides we get,

2 = e^{0.105t}

ln 2 = ln e ^{0.105t}

ln 2 = 0.105t (ln e)

ln 2 = 0.105t

Use the calculator to solve the logarithm values,

0.693147 = 0.105t

t = 6.666

Thus it takes 6.66 years to double the money.

**Ex 1:**

Find the decay constant k where time t = 4 days, N = 900 and No = 1000 and find N after 7 days?

**Sol:**

Let us take the decay formula

N = No . e ^{kt}

ln (N /No) = k . t

ln (900 / 1000) = k . 4

-0.105 = 4k

k = -0.105/4

k = -0.026 day^{ -1}

Thus we can calculate the Value of N after 7 days,

N = 1000 e ^{-0.026 × 7}

N = -999.818.

**Ex: 2**

Find the exponential growth factor if it takes $1500to double at 41/4 % compounded continuously

**Sol:**

First we have to take the formula,

A = Pe^{rt}

The rate value which can be taken as 0.105

3000 = 1500 e ^{0.125t}

We have to take natural log on both sides we get,

2 = e^{0.125t}

ln 2 = ln e ^{0.125t}

ln 2 = 0.125t (ln e)

ln 2 = 0.125t

Use the calculator to solve the logarithm values,

0.693147 = 0.125t

t = 5.54

Thus it takes 5.54 years to double the money.